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438 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

that is, the weights are inversely proportional to the variance of Ui or Yi conditional upon the given Xi, it being understood that var(ui | Xi) = var(Yi | Xi) = of.

Differentiating (2) with respect to p* and p2, we obtain

WiU2 dp 2

Setting the preceding expressions equal to zero, we obtain the following two normal equations:

Notice the similarity between these normal equations and the normal equations of the unweighted least squares.

Solving these equations simultaneously, we obtain

The variance of P2 shown in (11.3.9) can be obtained in the manner of the variance of p2 shown in Appendix 3A, Section 3A.3.

Note: Y* = J2 WiYi/JfWi and X = J2 WiXi/JfWi. As can be readily verified, these weighted means coincide with the usual or unweighted means Y and X when Wi = W, a constant, for all i.

11A.3 PROOF THAT E(S2) = ct2 IN THE PRESENCE OF HETEROSCEDASTICITY

Consider the two-variable model:

where varU) = of Now

0 2 = LU _ J2(Yi _ Yi )2 _ JJPi + p2Xi + Ui _ /Si _ p? Xi ]2

CHAPTER ELEVEN: HETEROSCEDASTICITY 439

Noting that (¡Y1 - ¡1) = -(- ¡*)X+u, and substituting this into (*) and taking expectations on both sides, we get:

E(a2) = {—e-2 var(Ä) + E [£(u — u)2]} 1 t e -fa2 + (n — 1) e a2

As you can see from (3), if there is homoscedasticity, that is, ai* = a* for each i, E(a *) = a *. Therefore, the expected value of the conventionally computed a * = E u*/(n - *) will not be equal to the true a * in the presence of heteroscedasticity.1

11A.4 WHITE'S ROBUST STANDARD ERRORS

To give you some idea about White's heteroscedasticity-corrected standard errors, consider the two-variable regression model:

Vx2a 2

Since a2 are not directly observable, White suggests using U2, the squared residual for each i, in place of a2 and estimate the var (¡32) as follows:

White has shown that (3) is a consistent estimator of (2), that is, as the sample size increases indefinitely, (3) converges to (2).2

Incidentally, note that if your software package does not contain White's robust standard error procedure, you can do it as shown in (3) by first running the usual OLS regression, obtaining the residuals from this regression and then using formula (3).

White's procedure can be generalized to the ^-variable regression model

1Further details can be obtained from Jan Kmenta, Elements of Econometrics, *d. ed., Macmillan, New York, 1986, pp. *76-*78.

*To be more precise, n times (3) converges in probability to E[(Xi - px) is the probability limit of n times (*), where n is the sample size, is the expected value of X, and aX is the (population) variance of X. For more details, see Jeffrey M. Wooldridge, Introductory Econometrics; A Modern Approach, South-Western Publishing, *000, p. *50.

440 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

The variance of any partial regression coefficient, say fy, is obtained as follows:

where U are the residuals obtained from the (original) regression (4) and wj are the residuals obtained from the (auxiliary) regression of the regressor Xj on the remaining regressors in (4).

Obviously, this is a time-consuming procedure, for you will have to estimate (5) for each X variable. Of course, all this labor can be avoided if you have a statistical package that does this routinely. Packages such as PcGive, Eviews, Microfit, Shazam, Stata, and Limdep now obtain White's heteroscedasticity-robust standard errors very easily.